Download Reverse Mathematics: Proofs from the Inside Out, by John Stillwell
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Reverse Mathematics: Proofs from the Inside Out, by John Stillwell
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Review
"If you are not familiar with this relatively new research about the foundations and and minimal assumptions needed to develop the massive mathematical structure, this provides a good informal guideline."---Adhemar Bultheel, European Mathematical Society"Reverse mathematics is the mathematical logician's version of zoology or botany, a classification of mathematical theorems in terms of the strength of the axioms needed to prove them. Stillwell carefully situates the field in the broader context of the history of mathematics and its foundations, and does a fine job of making the whole endeavor accessible to a general mathematical audience."--Jeremy Avigad, Carnegie Mellon University"Filling an important niche, this book gives readers a good picture of the basics of reverse mathematics while suggesting several directions for further reading and study. It provides a context for the questions investigated by reverse mathematics and makes an extended argument for their significance within contemporary mathematical practice."--Denis Hirschfeldt, University of Chicago"Reverse mathematics is a major research direction in the foundations of mathematics and mathematical logic, and the insights obtained from reverse mathematics will interest a wide mathematically minded audience. Stillwell's book is self-contained and includes much background material in analysis, mathematical logic, combinatorics, and computability. I heartily commend this very readable and accessible book."--Stephen Simpson, Vanderbilt University
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From the Back Cover
"Reverse mathematics is the mathematical logician's version of zoology or botany, a classification of mathematical theorems in terms of the strength of the axioms needed to prove them. Stillwell carefully situates the field in the broader context of the history of mathematics and its foundations, and does a fine job of making the whole endeavor accessible to a general mathematical audience."--Jeremy Avigad, Carnegie Mellon University"Filling an important niche, this book gives readers a good picture of the basics of reverse mathematics while suggesting several directions for further reading and study. It provides a context for the questions investigated by reverse mathematics and makes an extended argument for their significance within contemporary mathematical practice."--Denis Hirschfeldt, University of Chicago"Reverse mathematics is a major research direction in the foundations of mathematics and mathematical logic, and the insights obtained from reverse mathematics will interest a wide mathematically minded audience. Stillwell's book is self-contained and includes much background material in analysis, mathematical logic, combinatorics, and computability. I heartily commend this very readable and accessible book."--Stephen Simpson, Vanderbilt University
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Product details
Hardcover: 200 pages
Publisher: Princeton University Press (January 1, 2018)
Language: English
ISBN-10: 9780691177175
ISBN-13: 978-0691177175
ASIN: 0691177171
Product Dimensions:
6.4 x 1 x 9.3 inches
Shipping Weight: 1.1 pounds (View shipping rates and policies)
Average Customer Review:
4.2 out of 5 stars
5 customer reviews
Amazon Best Sellers Rank:
#191,872 in Books (See Top 100 in Books)
This is not a beginners book. But if you have been down the road before it is brilliantly written. It is mathematics, but not overbearing.The author covers numbers, analysis, computability, the key elements of current day mathematics. He does so in a very readable manner focusing on reverse theory, namely looking at the results to see the assumptions. Hilbert and his Geometry was to some degree to me a similar task a century ago. But instead of being ponderous the author is succinct, clear, and understanding. I found the analysis section and interesting interpretation having spent time decades ago in the area.For anyone interested in a different view of mathematics this is an idea addition and very well worth the read. As another reviewer wrote, the first chapter is easy, the other challenging but well worth the effort.
The author describes this new area very well. A great introductory book. He starts with natural numbers and gets all the way to reals and set theory. I like the way he transitions from set theory to Godel and computation.
The content is OK, but it is not formatted to read like other kindle books. In fact , you can't even read it on the Kindle. It is readable on the PC but even that is awkward and kind of "flaky" because the controls can give inconsistent results. If you are serious about this book, I would recommend the hard copy. The Kindle version is just very annoyingly awkward.
The overt prize for reading "Reverse Mathematics: Proofs from the Inside Out" is learning how to prove some of the most important theorems of analysis using the absolute barest set of axioms possible. More importantly, however, the reader gains an appreciation of why the field of reverse mathematics is so interesting and important. What does it mean for a theorem and an axiom to be equivalent within some base system? Is there a single underlying argument that is shared between all proofs that lie on the same axiomatic "level"? A prerequisite for answering questions like these is a basic understanding of elementary formal systems and of computability theory.Professor Stillwell provides all of this in a surprisingly self-contained book that is fun to read. He starts with an illuminating history of several upheavals in the foundations of mathematics that are analogous, and lead naturally to, what reverse mathematics is doing today. He then dives into the prerequisites I mentioned above. Computability turns out to be a natural base for the theorems of analysis and therefore much of the book is focused on teaching computability theory.The first chapter may give you the impression that you can quickly turn the pages, but make no mistake: this is a math book. Depending on your background level, you may have to put significant effort into understanding the concepts. For example, I had absolutely no knowledge of computability theory, so I had to do a lot of re-reading, and working out of concepts on my own, in the appropriate chapters. Likewise, if your knowledge of analysis is rusty, you may have to spend extra time on Chapter 3, where Stillwell provides the classical proofs of the theorems that will take center stage later. All the information is there however, making this book a great stepping-stone to more advanced treatments of reverse mathematics for someone with only an undergraduate level background in math.
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